We obtain characterizations of a variable version of Lipschitz spaces in terms of the boundedness of commutators of Calderon-Zygmund and fractional type operators in the context of the variable exponent Lebesgue spaces Lp(⋅), where the symbols of the commutators belong to the Lipschitz spaces. A useful tool is a pointwise estimate involving the sharp maximal operator of the commutator and certain associated maximal operators, which is new even in the classical context. Some boundedness properties of the commutators between Lebesgue and Lipschitz spaces in the variable context are also proved.

Characterization of Lipschitz Functions via the Commutators of Singular and Fractional Integral Operators in Variable Lebesgue Spaces

We define a Muckenhoup-type condition on weighted Morrey spaces using the Kothe dual of the space. We show that the condition is necessary and sufficient for the boundedness of the maximal operator defined with balls centered at the origin on weighted Morrey spaces. A modified condition characterizes the weighted inequalities for the Calderon operator. We also show that the Muckenhoup-type condition is necessary and sufficient for the boundedness on weighted local Morrey spaces of the usual Hardy–Littlewood maximal operator, simplifying the previous characterization of Nakamura–Sawano–Tanaka. For the same operator, in the case of global Morrey spaces the condition is necessary and for the sufficiency we add a local $$A_p$$
 condition. We can extrapolate from Lebesgue $$A_p$$
 -weighted inequalities to weighted global and local Morrey spaces in a very general setting, with applications to many operators.

Muckenhoupt-Type Conditions on Weighted Morrey Spaces

We define a Muckenhoup-type condition on weighted Morrey spaces using the Kothe dual of the space. We show that the condition is necessary and sufficient for the boundedness of the maximal operator defined with balls centered at the origin on weighted Morrey spaces. A modified condition characterizes the weighted inequalities for the Calderon operator. We also show that the Muckenhoup-type condition is necessary and sufficient for the boundedness on weighted local Morrey spaces of the usual Hardy-Littlewood maximal operator, simplifying the previous characterization of Nakamura-Sawano-Tanaka. For the same operator, in the case of global Morrey spaces the condition is necessary and for the sufficiency we add a local $A_p$ condition. We can extrapolate from Lebesgue $A_p$-weighted inequalities to weighted global and local Morrey spaces in a very general setting, with applications to many operators.

Muckenhoupt-type conditions on weighted Morrey spaces

The paper studies the heat kernel of the Schrodinger operator with magnetic fields and of uniformly elliptic operators with non-negative electric potentials in the reverse Holder class which includes non-negative polynomials as typical examples. The main aim of the paper is to give a pointwise estimate of the heat kernel of the operators above which is affected by magnetic fields and non-negative degenerate electric potentials. A weighted smoothing estimate for the semigroup generated by the operators above is also given.

An Estimate on the Heat Kernel of Magnetic Schrodinger Operators and Uniformly Elliptic Operators with Non-negative Potentials (Harmonic Analysis and Nonlinear Partial Differential Equations)

Extrapolation and weighted norm inequalities between Lebesgue and Lipschitz spaces in the variable exponent context

For a proper open set $$\Omega $$
 immersed in a metric space with the weak homogeneity property, and given a measure $$\mu $$
 doubling on a certain family of balls lying “well inside” of $$\Omega $$
 , we introduce a local maximal function and characterize the weights $$w$$
 for which it is bounded on $$L^p(\Omega ,w d\mu )$$
 when $$1<p<\infty $$
 and of weak type $$(1,1)$$
 . We generalize previous known results and we also present an application to interior Sobolev’s type estimates for appropriate solutions of the differential equation $$\Delta ^m u=f$$
 , satisfied in an open proper subset $$\Omega $$
 of $$\mathbb R ^n$$
 . Here, the data $$f$$
 belongs to some weighted $$L^p$$
 space that could allow functions to increase polynomially when approaching the boundary of $$\Omega $$
 .

Local maximal function and weights in a general setting

Lipschitz type smoothness of the fractional integral on variable exponent spaces

For a proper open set Ω immersed in a metric space with the weak homogeneity property, and given a measure μ doubling on a certain family of balls lying “well inside” of Ω, we introduce local operators of singular and fractional type and study their boundedness properties on weighted Lp(Ω), 1 ≤ p < ∞, for weights in local Muckenhoupt classes.

Local fractional and singular integrals on open subsets

In this work we search for boundedness results for operators related to the semigroup generated by the degenerate Schrodinger operator ${{\mathscr{L}}} u = -\frac {1}{\omega } \text {div} A\cdot \nabla u +V u$
 , where ω is a weight, A is a matrix depending on x and satisfying λω(x)|ξ|2 ≤ A(x)ξ ⋅ ξ ≤Λω(x)|ξ|2 for some positive constants λ, Λ and all x, ξ in $\mathbb {R}^{d}$
 , assuming further suitable properties on the weight ω and on the non-negative potential V. In particular, we analyze the behaviour of T∗, the maximal semigroup operator, ${{\mathscr{L}}}^{-\alpha /2}$
 , the negative powers of ${{\mathscr{L}}}$
 , and the mixed operators ${{\mathscr{L}}}^{-\alpha /2}V^{\sigma /2}$
 with 0 < σ ≤ α on appropriate functions spaces measuring size and regularity. As in the non degenerate case, i.e. ω ≡ 1, we achieve these results by first studying the case V = 0, obtaining also some boundedness properties in this context that we believe are new.

Boundedness of Operators Related to a Degenerate Schrödinger Semigroup

We study singular integral operators with Hilbert-valued kernels in the setting of Rn with non-necessarily doubling measures. We obtain an explicit formula for these operators following a similar approach as in Macias et al. (Adv Math 93:25–60, 1992). By using this formula and a result due to Krein we get a T1-theorem in this context. Finally, we develop a theory for antisymmetric kernels and we apply the results to the oscillation operators related to the Riesz transform.

Beatriz Eleonora Viviani

Papers

Local maximal function and weights in a general setting

Lipschitz type smoothness of the fractional integral on variable exponent spaces

Local fractional and singular integrals on open subsets

Boundedness of Operators Related to a Degenerate Schrödinger Semigroup

An explicit expression for singular integral operators with non-necessarily doubling measures